Worked examples
The question
Question: estimate √20.
√20
The square root of 20 is the number that times itself makes 20.
√20 × √20 = 20
Careful!
The square root is not half.
10 times 10 is 100, far more than 20.
Careful: not √20 = 10 ✗
1 List the squares
First, the square numbers near 20.
3 times 3 is 9.
4 times 4 is 16.
5 times 5 is 25.
6 times 6 is 36.
2 Find where it fits
20 is between the squares 16 and 25.
= 16 < 20 < 25
A bigger number has a bigger square root.
So √20 is between √16 and √25.
= √16 < √20 < √25
√16 is 4.
√25 is 5.
So √20 is between 4 and 5.
= 4 < √20 < 5
3 Which is closer
20 is 4 more than 16, and 5 less than 25.
= 20 − 16 = 4, 25 − 20 = 5
4 is less.
So 20 is closer to 16, and √20 is closer to 4.
4 < 5: closer to 16
So √20 is between 4 and 5, closer to 4.
= 4 < √20 < 5, closer to 4
4 Check
Let's check with the number halfway: 4.5.
4.5 times 4.5 is 20.25, more than 20.
4.5 × 4.5 = 20.25 > 20
So √20 is below 4.5: closer to 4.
The answer is right!
√20 < 4.5: closer to 4
Let's look back.
First: list the squares.
Then: find where it fits.
Then: which is closer.
Last: check.
Answer = 4 < √20 < 5, closer to 4
▶ Watch it solved
The question
Question: estimate √90.
√90
The square root of 90 is the number that times itself makes 90.
√90 × √90 = 90
Careful!
The square root is not half.
45 times 45 is 2025, far more than 90.
Careful: not √90 = 45 ✗
1 List the squares
First, the square numbers near 90.
8 times 8 is 64.
9 times 9 is 81.
10 times 10 is 100.
11 times 11 is 121.
2 Find where it fits
90 is between the squares 81 and 100.
= 81 < 90 < 100
A bigger number has a bigger square root.
So √90 is between √81 and √100.
= √81 < √90 < √100
√81 is 9.
√100 is 10.
So √90 is between 9 and 10.
= 9 < √90 < 10
3 Which is closer
90 is 9 more than 81, and 10 less than 100.
= 90 − 81 = 9, 100 − 90 = 10
9 is less.
So 90 is closer to 81, and √90 is closer to 9.
9 < 10: closer to 81
So √90 is between 9 and 10, closer to 9.
= 9 < √90 < 10, closer to 9
4 Check
Let's check with the number halfway: 9.5.
9.5 times 9.5 is 90.25, more than 90.
9.5 × 9.5 = 90.25 > 90
So √90 is below 9.5: closer to 9.
The answer is right!
√90 < 9.5: closer to 9
Let's look back.
First: list the squares.
Then: find where it fits.
Then: which is closer.
Last: check.
Answer = 9 < √90 < 10, closer to 9
▶ Watch it solved
The question
Question: estimate √10 to the nearest tenth.
√10
The square root of 10 is the number that times itself makes 10.
√10 × √10 = 10
Careful!
The square root is not half.
5 times 5 is 25, far more than 10.
Careful: not √10 = 5 ✗
1 List the squares
First, the square numbers near 10.
2 times 2 is 4.
3 times 3 is 9.
4 times 4 is 16.
5 times 5 is 25.
2 Find where it fits
10 is between the squares 9 and 16.
= 9 < 10 < 16
A bigger number has a bigger square root.
So √10 is between √9 and √16.
= √9 < √10 < √16
√9 is 3.
√16 is 4.
So √10 is between 3 and 4.
= 3 < √10 < 4
3 Which is closer
10 is 1 more than 9, and 6 less than 16.
= 10 − 9 = 1, 16 − 10 = 6
1 is less.
So 10 is closer to 9, and √10 is closer to 3.
1 < 6: closer to 9
4 Try tenths
√10 is a bit more than 3.
So we try 3.1, 3.2 and so on, and square each one.
3.1 times 3.1 is 9.61.
That is less than 10: too small.
= 3.1 × 3.1 = 9.61 → too small
3.2 times 3.2 is 10.24.
That is more than 10: too big.
= 3.2 × 3.2 = 10.24 → too big
So √10 is between 3.1 and 3.2.
= 3.1 < √10 < 3.2
5 Pick the nearest tenth
10 is 0.39 more than 9.61, and 0.24 less than 10.24.
So it is closer to 10.24.
= 10 − 9.61 = 0.39, 10.24 − 10 = 0.24
0.24 < 0.39: closer to 3.2
So √10 is about 3.2, to the nearest tenth.
= √10 ≈ 3.2
6 Check
Let's check with the number halfway: 3.15.
3.15 times 3.15 is 9.9225, less than 10.
3.15 × 3.15 = 9.9225 < 10
So √10 is above 3.15.
To the nearest tenth, it is 3.2.
The answer is right!
√10 > 3.15: nearest tenth 3.2
Let's look back.
First: list the squares.
Then: find where it fits.
Then: which is closer.
Then: try tenths.
Then: pick the nearest tenth.
Last: check.
▶ Watch it solved
